Given that $y = \frac{1}{4 + \sqrt{(x - 1)}}$, complete the table below with values of $y$ corresponding to $x = 3$ and $x = 5$ - Edexcel - A-Level Maths Pure - Question 1 - 2010 - Paper 6

Using the trapezium rule:

The estimate for $I$ is given by:

$I \approx \frac{1}{2} \times (0.2 + 0.1847 + 0.1745 + 0.1667) \times (5 - 2) = \frac{1}{2} \times (0.7259) \times 3 \approx 1.08885 \Rightarrow 0.543$

To find the exact value of $I$ using the substitution, let: $x = u - 4 + 1 = u - 3$ then: $dx = du$

Now substitute into the integral:

$I = \int_{2}^{5} \frac{1}{4 + \sqrt{(x - 1)}}dx = \int_{2}^{5} \frac{1}{4 + \sqrt{(u - 3 - 1)}}du$

Performing the integration leads us to the exact value of $I = 2 + 8 \ln \left(\frac{5}{6}\right) + \frac{1}{6}\approx 2.3754$.